Dribus Thesis Slides

Preliminaries Background Green-Griffiths Our Approach Tangent Groups Coniveau Machine Research Program References

Infinitesimal Theory of Chow Groups

Benjamin F. DribusAdvisor: J.W. Hoffman

Louisiana State University

March 7, 2014


Introduction

• The Chow groups ChpX for a smooth algebraic variety X over

a field k are central to algebraic geometry.

• Poorly understood for p ≥ 2.

• Green-Griffiths 2005 [1]: “Study their tangent groups!”

• Dribus-Hoffman-Yang have improved this approach:• New infinitesimal theory of Chp

X .• Generalized tangent groups TY Chp

X .

• Depends on:• Bass-Thomason algebraic K -theory.• Negative cyclic homology.• Coniveau spectral sequence.• Algebraic Chern character.


Notation and Conventions

• Distinguished technical terms are blue: Chow groups.

• Math symbols are red: ChpX .

• New concepts and results are green: coniveau machine.

• Reference hyperlinks are grey: Bloch 1972 [8].

• Parentheses are often suppressed: ChpX , not Chp(X ).


Where We’re Headed I

Overarching goal: define computable invariants involvingalgebraic cycles, and show how to compute them!

• Coniveau machine: special diagram of functors and naturaltransformations defined for this purpose.

• Theorem (DHY, 2012): Coniveau machine for Chow groupsexists.

• Corollary: Can compute generalized tangent groups of Chowgroups via negative cyclic homology:

TY ChpX ≅ Hp

Zar(X ,HNp,X×kY ,Y ).


Where We’re Headed II

• Schematic diagram of coniveau machine:

Cousin

resolution

of absoluteK -theory

Kp,X

1

splitinclusion

Cousin

resolution

of augmentedK -theory

Kp,X×kY

2

splitprojection

Cousin

resolution

of relativeK -theory

Kp,X×kY ,Y

3relativeChern

characterCousin

resolutionof relativenegative

cyclic

homologyHNp,X×kY ,Y

4

generalized tangent map

• Generalized tangent groups TY ChpX computed via fourth

column.

• Generalized tangent map takes “deformations” to their“tangent elements.”


Algebraic Varieties

• Technically, an algebraic variety X is an integral, separatedscheme of finite type over an algebraically closed field k .

• Informally, X defined locally by vanishing of polynomials.

• Example: X the complex algebraic curve y 2 = x3 + x2 − x + 1:

real points of X X as a Riemann surface

x

y


Algebraic Cycles• Codimension-p algebraic cycle z on X :

• Formal linear combination of codimension-p subvarieties of X .• Written as a finite formal sum:

z = ∑x∈ZarpX

nxx .

• Subvarieties are labeled by their generic points x .• ZarpX : set of codimension-p points in Zariski topology on X .• The nx are multiplicities, usually integers.

• Example: codimension-one cycle on an algebraic curve X :

+2

+1

−1

−4

X


Cycle Groups

• ZpX : set of codimension-p cycles on X .• Z p

X an abelian group called pth cycle group of X .• Group operation on Z p

X induced by addition of multiplicities.

• Example: addition of codimension-one cycles on a curve:

X−1+2

−1

+1

−1

+1

−1

X

−2

+1

−1

+2X


Rational Equivalence

• Cycle groups are “huge,” and have poor intersection theory.

• Strategy: impose an adequate equivalence relation.• Introduced by Samuel 1958 [2].• Rational equivalence: two cycles “belong to a P1-family.”

Xz1z2

P1

λ1

λ2

V

X × λ1

X × λ2

• Group of rational equivalence classes of codimension-p cyclesis the pth Chow group Chp

X of X .


“Easy Case:” Divisors and Ch1X

• Divisors: codimension-1 cycles.

• Examples: points on a curve, curves on a surface.

• “Easy” because:• Ch1

X ≅ PicX (Picard group, an algebraic group.)• Algebraic and homological equivalence coincide.• Gives Picard variety Pic0

X , an abelian variety.• Example: X ≅ Pic0

X for genus-1 curve X , via Abel-Jacobi map.

• Lefschetz (1,1)-theorem:• H2(X ,Z) ∩H1,1(X ) consists of divisor classes.• Special case of Hodge Conjecture.


ChpX for Higher Codimension

• For p ≥ 2, ChpX remains poorly understood.

• Famous results make situation look worse, not better:• Mumford 1969 [3]:

dim Ch2X =∞ (points on a surface)

• Griffiths 1969 [4]:

alg. equ. ≠ hom. equ. (curves on a threefold)

• Clemens 1983 [5]:

dimalg. equ.

hom. equ.=∞ (codimension 2)

• Conjectures: Hodge conjecture, Tate conjecture, conjecturedfiltrations of ChpX , etc.


Approach of Green and Griffiths

• Strategy: “Linearize the problem!”• Conceptually similar to Lie theory: linearize Lie group to

obtain Lie algebra.• Uses algebraic K -theory.

• Previous/contemporary related work:• Van der Kallen [6]: tangent space of K2.• Bloch [7], [8]: Ch2

X via K -theory; tangent space of Ch2X .

• Quillen [9]: extensions via higher K -theory.• Stienstra [10], [11], [12]: Cartier-Dieudonne theory for Chow

groups (very sophisticated)• Hesselholt [13]: K -theory of truncated polynomial algebras.• Many others!


Bloch’s Formula

• Bloch-Quillen 1972 [7], [9]:

ChpX ≅ Hp

Zar(X ,Kp,X ).

(Zariski sheaf cohomology.)

• Computed via following sheafified Cousin complex:

0 Kp,X ∐

x∈Zar0X

Kp(kx) ∐

x∈Zar1X

Kp−1(kx) ∐x∈ZarpX

K0(kx) 0

• Called Bloch-Gersten-Quillen resolution (BGQR) of Kp,X .• Milnor version: use Milnor K -theory KM

p,X instead.

(Kerz 2006 [14]; previously known only up to torsion!)


Green and Griffiths’ Tangent Group TGGChpX

• Green and Griffiths 2005 [1] define tangent group at theidentity of Chp

X via Milnor K -theory:

TGGChpX ∶= Hp

Zar(X ,TKMp,X ) = Hp

Zar(X ,Ωp−1X /Q).

• TKMp,X : tangent sheaf of KM

p,X .

• Ωp−1X/Q: sheaf of absolute Kahler differentials.

• TKMp,X ≅ Ωp−1

X/Q: primitive relative algebraic Chern character.

• TGGChpX : my notation.

• Brings arithmetic considerations to the forefront, even forcomplex varieties.


Tangent Sequences and Sheaves

• Green-Griffiths’ focus: X a smooth algebraic surface.

• Identify the tangent sequence to the BGQR of K2,X as thefollowing sheafified Cousin complex:

0 Ω1X /Q ∐

x∈Zar0X

H0x (Ω

1X /Q) ∐

x∈Zar1X

H1x (Ω

1X /Q) ∐

x∈Zar2X

H2x (Ω

1X /Q) 0

• Called Cousin flasque resolution (CFR) of Ω1X/Q.

• Hqx (Ω

1X/Q) is local cohomology (Hartshorne 1966 [16]).


Computing Ch2X and TGGCh2

X

• Consider the final maps d1 and Td1 in the BGQR of KX ,2 andthe CFR of Ω1

X /Q:

∐

x∈Zar1X

K1(kx) ∐

x∈Zar2X

K0(kx) 0d1

∐

x∈Zar1X

H1x (Ω

1X /Q) ∐

x∈Zar2X

H2x (Ω

1X /Q) 0

Td1

• By the definition of sheaf cohomology:

Ch2X ≅ H2

Zar(X ,K2,X ) =Γ

Γ

∐

x∈Zar2X

K0(kx)

Im d1

• Similarly, using Green-Griffiths’ definition:

TGGCh2X = H2

Zar(X ,Ω1X /Q) =

Γ

Γ

∐

x∈Zar2X

H2x (Ω

1X /Q)

Im Td1


Geometric Interpretation

• Green-Griffiths provide geometric interpretation for T Ch2X in

terms of “arcs” of zero cycles and arcs of rationalequivalences.

• “Arcs” are defined only informally.

• The idea is familiar from differential geometry and Lie theory:

0G

e

g

arc at g in Lie group Gwith tangent

G

e

g

transported toidentity e


Improving on GG

GG’s work is presented as exploratory, not definitive.

• Conceptual framework is unclear.

• Rigor is lacking.

• Difficult to apply, or even assess.

• Better tools available: Bass-Thomason K -theory, cyclichomology, algebraic Chern character, etc.

• Recent results offer improvements (e.g. Kerz 2006 [14],Cortinas et al. 2008 [15]).

• Vast generalizations are possible.


Introducing Our Approach

Our approach is based on four general ideas:

1. Cohomology theories with supports (CTS).

2. Coniveau filtration of topological spaces, and the resultingconiveau spectral sequences (CSS).

3. Nilpotent augmentation.

4. Exponential and logarithmic-type maps.


Cohomology Theories with Supports

A cohomology theory with supports (CSS) is a special family offunctors.

• Source: distinguished category S of topological spaces (e.g.,separated schemes over a field k).

• Target: abelian category A.

• Respects topological structure in a special way.

• Often arises from a spectrum-valued functor (substratum).

• Examples:• Bass-Thomason algebraic K -theory via substratum K.• Negative cyclic homology via substratum HN.• Many others (Colliot-Thelene, Hoobler, and Kahn 1997 [17]).

• Non-examples: Quillen K -theory, Milnor K -theory.


Coniveau Filtration and Coniveau Spectral Sequence

• Coniveau filtration organizes the points of a topological spaceby codimension.

• “Coniveau” means “codimension” (French).• Example: Zariski topology on n-dimensional scheme:

⊘ ⊂ Zar≥nX ⊂ Zar≥n−1X ⊂ ... ⊂ Zar≥1

X ⊂ Zar≥0X = ZarX

• Very general method.

• Coniveau spectral sequence (CSS) for a CTS on a category S:• Induced by coniveau filtration via exact couple.• Rows are Cousin complexes.• Examples: BGQR of K2,X and CFR of Ω1

X/Q are sheafifiedversions of Cousin complexes.


Nilpotent Augmentation

• Algebraic notion of “infinitesimal degrees of freedom.”

• Simplest example: R ↦ Rε ∶= R[ε]/ε2.• Idea: ε is “so small that ε2 = 0.”• If R is a k-algebra, then Rε = R ⊗k kε.• kε: “algebra of dual numbers.”• For k-schemes: X ↦ X ×k kε (fiber product).• First-order infinitesimal theory; gives tangent space.

• More general: replace kε:• Use any k-algebra A such that Ker[A→ k] is nilpotent.• For k-schemes: X ↦ X ×k Y , where Y = Spec A.• Higher-order theory; gives generalized tangent space.

• Does not affect topological structure!


Exponential and Logarithmic-Type Maps

• Ubiquitous in mathematics; e.g., Lie theory.

• Algebraic setting: algebraic Chern character.• Primitive version: dlog map from Milnor K -theory to

differential forms.• Maps algebraic K -theory to negative cyclic homology.• May be viewed as natural transformation of functors:

ch ∶ K↦ HN.

• Relative version induces isomorphisms for nilpotentaugmentations:

chp ∶Kp,X×kY ,Y →HNp,X×kY ,Y

• Related to Goodwillie’s isomorphism [18] (sheafified):

ρp ∶Kn,X×kY ,Y ⊗Q→HCp−1,X×kY ,Y ⊗Q


Our Approach to Chow Groups

• Use Bass-Thomason K -theory, not Milnor K -theory.• View X ↦ Hp

Zar(X ,Kp,X ) as extension of Chow functor.

• Treat smooth and nilpotent-augmented structure functoriallyon same footing (Bass-Thomason K -theory essential).

• View BGQR of Kp,X as −pth sheafified Cousin complex fromCSS for K.

• Identify tangent sequence of BGQR as CFR of relativenegative cyclic homology HNp,X×kY ,Y .

• Define tangent map via relative algebraic Chern character:

chp,X×kY ,Y ∶Kp,X×kY ,Y →HNp,X×kY ,Y .


Generalizing Our Approach I

• Whenever a family of functors H = Hnn∈Z sufficientlyrespects topological structure, Chow group analogues exist.

• H must:• Take values in an abelian category.• Produce long exact sequences for inclusions of closed subsets.• Satisfy an appropriate local condition (effacement).

• Such a family H is called an effaceable cohomology theorywith supports (CTS).

• K and HN are examples.• Colliot-Thelene, Hoobler, and Kahn 1997 [17] mention others.• The CSS of an effaceable CTS H yields CFR’s of the

associated sheaves HpX .


Generalizing Our Approach II

• To admit nilpotent augmentation, a CTS H must be mediatedby ring structure.

• Example: H is Bass-Thomason algebraic K -theory:• Kp,X is the sheaf associated to the presheaf U ↦ Kp,OU

.• OU is the ring of sections of OX over U.• Nilpotent augmentation occurs at level of OU .• E.g., for schemes over a field k , OU ↦ OU ⊗k A, where A is

generated by nilpotents.

• Hence, ringed spaces are the natural setting for nilpotentstructure, though the CSS is purely topological.

• Generalized tangent maps are natural transformations fromaugmented to relative CTS’s.


Extending the Chow Functors

• Chow functor X ↦ ChpX is purely topological.

• Must extend it to “see nilpotent structure.”

• Two possible choices:

1. Bloch-Milnor: X ↦ Hp

Zar(X ,KM

p,X ).

2. Bloch: X ↦ Hp

Zar(X ,Kp,X ).

• Same for X smooth.

• Different in general!


Defining Tangent Groups

• Goal: define tangent groups at the identity of ChpX .• Analogous to Lie algebras.• Depends on choice of extension of Chow functor.

• Green-Griffiths [1] use Bloch-Milnor to define tangent group:

TGGChpX ∶= Hp

Zar(X ,TKMp,X ).

• We (Dribus-Hoffman-Yang) use Bloch to define generalizedtangent groups:

TY ChpX ∶= Hp

Zar(X ,Kp,X×kY ,Y ).


Why Our Definition of TY ChpX ?

• Algebraic K -theory decomposes into eigenspaces of Adamsoperations:

Kp,X =p

⊕i=0

K(i)p,X .

• Milnor K -theory KMp,X involves only pth eigenspace K

(p)p,X .

• Example: TK3,X ≅ Ω2X/Q ⊕OX but TKM

3,X ≅ Ω2X/Q.

Missing OX summand!

• Hence, Green-Griffiths tangent group TGGChpX neglects the

other eigenspaces!

• TY ChpX can “see” all the eigenspaces.

• Also allows arbitrary nilpotent structure (not just first order)!


Computing the Tangent Groups

• Can we compute the groups TY ChpX ? If so, how?

• Recall overarching goal: define computable invariantsinvolving algebraic cycles, and show how to compute them!

• Computationally:• Cycle groups: hard.• Chow groups: hard.• Algebraic K -theory: hard.• Cyclic homology: relatively easy.• Differential forms: easy.

• Strategy: use Chern character to convert “hard” K -theoryinto “easy” cyclic homology or differential forms.

• Examples:TKM

p,X ≅ Ωp−1X /Q.

Kp,X×kY ,Y ≅HNp,X×kY ,Y .


Method: Coniveau Machine

• Coniveau machine converts information about TY ChpX into

information about negative cyclic homology.

• Schematic diagram:

Cousin

resolution

of absoluteK -theory

Kp,X

1

splitinclusion

Cousin

resolution

of augmentedK -theory

Kp,X×kY

2

splitprojection

Cousin

resolution

of relativeK -theory

Kp,X×kY ,Y

3relativeChern

characterCousin

resolutionof relativenegative

cyclic

homologyHNp,X×kY ,Y

4

generalized tangent map

• Nontrivial: The columns actually exist and fit together asshown!


Existence of Coniveau Machine: Early Versions

• Green-Griffiths [1]:• First-order theory, characteristic zero.• Focus on curves and surfaces.• Milnor K -theory, differential forms.• Define “tangent sequence to BGQR” for a surface.• Use different terminology.

• Sen Yang [19]:• First-order theory, characteristic zero.• Arbitrary dimensions• Also worked on typical curves (?)• Different terminology.

• Dribus-Hoffman-Yang upcoming [20].


Existence of the Machine: Background

General version (for Chow groups) requires:

• CSS: Grothendieck-Hartshorne circa 1960 [16].

• Bloch’s formula: Bloch-Quillen, by 1972 [7], [9].

• Effaceability of Bass-Thomason K -theory: Colliot-Thelene,Hoobler, and Kahn 1997 [17]; implicit in Thomason 1990 [21].

• Effaceability of negative cyclic homology: implicit inWeibel 1991 [22] and Keller 1998 [23].

• Relative algebraic Chern character an isomorphism:(nilpotent case) Cortinas et al. 2008 [15].


Column 1: Cousin Resolution of Kp,X

• Standing assumptions:• S a “suitable” category of schemes over a field k.• X in S smooth algebraic variety.• Y in S fixed separated scheme.

• To show: this resolution exists!

• Easy: this is just the BGQR via the CSS:

0 Kp,X ∐

x∈Zar0X

Kp(kx) ∐

x∈Zar1X

Kp−1(kx) ∐x∈ZarpX

K0(kx) 0

• Requires Quillen’s devissage to use residue fields kx ; appliesonly in smooth case!

• More general Bass-Thomason form using supports:

0 Kp,X ∐

x∈Zar0X

Kp,X on x ∐

x∈Zar1X

Kp−1,X on x ∐x∈ZarpX

K0,X on x 0


Columns 2 and 3: Multiplying by Y

• To show: Cousin resolutions of Kp,X×kY and Kp,X×kY ,Y exist.

• Problem: scheme X ×k Y not smooth!

• Solution: define a new substratum:

X ↦ KYX ∶= KX×kY .

• Bass-Thomason form applies directly for KYp,X =Kp,X×kY :

0 KYp,X

∐

x∈Zar0X

KYp,X on x ∐

x∈Zar1X

KYp−1,X on x ∐

x∈ZarpX

KY0,X on x 0

• Follows Colliot-Thelene, Hoobler, and Kahn’s idea [17]: “newtheories out of old.”

• For Kp,X×kY ,Y , take fiber KYX ↦ KX ; gives third column and

maps for first three columns (details!)


Column 4: Relative Algebraic Chern Character

• To show:1. Cousin resolution of HNp,X×kY ,Y exists.2. Cousin resolutions of Kp,X×kY ,Y and HNp,X×kY ,Y are

isomorphic as complexes.

• First part follows as for K, since HN defines effaceable CSS.

• Second part requires X ↦ X ×k Y to be nilpotent.

• Then follows from fact that relative algebraic Chern characteris isomorphism:

chn,X×kY ,Y ∶Kp,X×kY ,Y →HNp,X×kY ,Y .

• This completes the coniveau machine!


Theorem and Corollary

• Theorem (DHY, 2012): The coniveau machine for Chowgroups exists. That is, given a nilpotent augmentationX ↦ X ×k Y of a smooth algebraic variety X over a field k,the −pth rows of the coniveau spectral sequences for absolute,augmented, and relative Bass-Thomason algebraic K -theoryand negative cyclic homology sheafify to yield flasqueresolutions of the corresponding sheaves on X , and therelative algebraic Chern character induces a functorialisomorphism between the resolutions of relative K -theory andrelative negative cyclic homology.

• Corollary: The generalized tangent groups TY ChpX of the

Chow groups T ChpX may be computed via negative cyclic

homology in the nilpotent case:

TY ChpX ≅ Hp

Zar(X ,HNp,X×kY ,Y ).


Generalized Coniveau Machine

• Coniveau machine generalizes to “commutative diagram offunctors and natural transformations with exact rows:”

EHrel,SEHaug,S EH,S

i j

EH+rel

,S EH+aug,S EH+,Si+ j+

chrel ∼ chaug ch(1.1.1.3)

• Here:• S: category of spaces; e.g., schemes over a field k.• H: CTS on S; e.g., algebraic K -theory.• H+: “additive version” of H; e.g., negative cyclic homology.• EH,S etc.: functors from spaces to CSS’s.• ch etc.: “logarithmic-type transformations;” e.g., algebraic

Chern character.


Future Directions I

• Compute new invariants:• TK3,X ≅ Ω2

X/Q ⊕OX ; look in OX piece!• Higher-order deformations.

• Symbolic K -theory:• My paper A Goodwillie-type Theorem for Milnor K -Theory.• Favorable remarks from Van der Kallen, Hesselholt, Stienstra.• Van der Kallen, Hesselholt: “generalize via de Rham Witt

theory.”• Stienstra: “combine with Cartier-Dieudonne theory.”• Loday symbols: reach other Adams eigenspaces!

• Geometric interpretations a la Green-Griffiths.• “Formal” versus “geometric” objects.• Infinitesimal existence results.


Future Directions II

• Infinitesimal filtrations of ChpX :

• Related to Beilinson’s conjectured filtrations of ChpX .

• Green-Griffiths version via differential forms.• Immediate reconsideration via cyclic homology.

• Abelian sums and residues.

• Positive characteristic. Much of our existing theory applies,but there are important new issues.

• Higher Chow groups.

• Other effaceable CTS’s. Colliot-Thelene, Hoobler, and Kahn’sidea [17] list a dozen; more have recently come to light.


THANKS!


References I

Mark Green and Phillip Griffiths.On the Tangent Space to the Space of Algebraic Cycles on aSmooth Algebraic Variety.Number 157 in Annals of Mathematics Studies. PrincetonUniversity Press, 2005.

Pierre Samuel.

Relations d’Equivalence en Geometrie Algebrique.Proceedings ICM 1958, 470-487, 1960.

David Mumford.Rational equivalence of 0-cycles on surfaces.J. Math. Kyoto Univ, 9(2):195–204, 1969.

Phillip Griffiths.On the Periods of Certain Rational Integrals, I and II.Annals of Mathematics, 90(3), 1969.


References II

Herbert ClemensGersten’s Conjecture and the homology of schemes.Publications mathematiques de l’I. H. E. S., 58:19–38, 1983.

Wilberd Van der Kallen.Le K2 des nombres duaux.Comptes Rendus de l’Academie des Sciences Paris, Serie A,273, pp. 1204-1207, 1971.

Spencer Bloch.K2 and Algebraic Cycles.Annals of Mathematics, 99, 2, pp. 349-379, 1974.

Spencer Bloch.On the Tangent Space to Quillen K -Theory.Lecture Notes in Mathematics, 341, pp. 205-210, 1972.


References III

Daniel Quillen.Higher algebraic K -theory I.Lecture Notes in Mathematics, 341, pp. 85-147, 1972.

Jan Stienstra.On the formal completion of the Chow group CH2(X ) for asmooth projective surface of characteristic 0.Indagationes Mathematicae (Proceedings), 86, 3, pp. 361-382,1983.

Jan Stienstra.Cartier-Dieudonn’e theory for Chow groups.Journal fur die reine und angewandte Mathematik, 355, pp.1-66, 1984.


References IV

Jan Stienstra.Correction to ‘Cartier-Dieudonn’e theory for Chow groups’.Journal fur die reine und angewandte Mathematik, 362, pp.218-220, 1985.

Lars Hesselholt.K -Theory of truncated polynomial algebras.Handbook of K -Theory, 1, 3, pp. 71-110, Springer-Verlag,Berlin, 2005.

Moritz Kerz.The Gersten Conjecture for Milnor K -Theory, 2006.Preprint.


References V

Weibel et al.Infinitesimal cohomology and the Chern character to negativecyclic homology.Mathematische Annalen, 2008.

Robin Hartshorne.Residues and Duality.Lecture Notes in Mathematics, 20, Springer-Verlag, 1966.

Jean-Louis Colliot-Thelene, Raymond T. Hoobler, and BrunoKahn.The Bloch-Ogus-Gabber Theorem,Fields Institute Communications, 16, pp. 31-94, 1997.

Thomas G. Goodwillie.Relative algebraic K -theory and cyclic homology.Annals of Mathematics, 124, 2, pp. 347-402, 1986.


References VI

Sen Yang.Higher Algebraic K -Theory and Tangent Spaces to ChowGroups.Thesis, 2013.

Benjamin F. Dribus, J. W. Hoffman, and Sen Yang.Infinitesimal Structure of Chow Groups and AlgebraicK -Theory.In preparation, 2014.

Robert W. Thomason.Higher Algebraic K -Theory of Schemes.The Grothendieck Festschrift III, Progress in Mathematics, 88,pp. 247-435, 1990.


References VII

Charles Weibel.Cyclic Homology for Schemes,Proceedings of the American Mathematical Society, 124, 6,pp. 1655-1662, 1996. Preprint URL:http://www.math.uiuc.edu/K-theory/0043/weibel.pdf

Bernhard Keller.On the Cyclic Homology of Ringed Spaces and Schemes, 1998Preprint; URL:http://www.math.uiuc.edu/K-theory/0259/crs.pdf

Benjamin F. Dribus.A Goodwillie-type Theorem for Milnor K -Theory.Preprint, to appear.

http://www.math.uiuc.edu/K-theory/0043/weibel.pdf

http://www.math.uiuc.edu/K-theory/0259/crs.pdf

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